1st Edition
A Functorial Model Theory Newer Applications to Algebraic Topology, Descriptive Sets, and Computing Categories Topos
This book is an introduction to a functorial model theory based on infinitary language categories. The author introduces the properties and foundation of these categories before developing a model theory for functors starting with a countable fragment of an infinitary language. He also presents a new technique for generating generic models with categories by inventing infinite language categories and functorial model theory. In addition, the book covers string models, limit models, and functorial models.
Introduction
Categorical Preliminaries
Categories and Functors
Morphisms
Functors
Categorical Products
Natural Transformations
Products on Models
Preservation of Limits
Model Theory and Topoi
More on Universal Constructions
Chapter Exercises
Infinite Language Categories
Basics
Limits and Infinitary Languages
Generic Functors and Language String Models
Functorial Morphic Ordered Structure Models
Chapter Exercises
Functorial Morphic Ordered Structure Models
Functorial Fragment Model Theory
Introduction
Generic Functors and Language String Models
Functorial Models As ¿-Chains
Models Glimpses From Functors
Structure Products
Higher Stratified Consistency and Completeness
Fragment Positive Omitting Type Algebras
Omitting Types and Realizability
Positive Categories and Consistency Models
More on Fragment Consistency
Chapter Exercises
Algebraic Theories, Categories, and Models
Ultraproducts on Algebras
Ultraproducts and Ultrafilters
Ultraproduct Applications to Horn Categories
Algebraic Theories and Topos Models
Free Theories and Factor Theories
T-Algebras and Adjunctions
Theory Morphisms, Products and Co-products
Algebras and the Category of Algebraic Theories
Initial Algebraic Theories and Computable Trees
Chapter Exercises
Generic Functorial Models and Topos
Elementary Topoi
Generic Functorial Models
Generic Functors
Initial D<A,G> Models
Positive Forcing Models
Functors Computing Hasse Diagram Models
Fragment Consistent Models
Homotopy theory of topos
Filtered colimits and comma categories
More on Yoneda Lemma
Chapter Exercises
Models, Sheaves, and Topos
PreSheaves
Duality, Fragment Models, and Topology
Duality
Lifts on Topos Models on Cardinalities
Chapter Exercises
Functors on Fields
Introduction
Basic Models
Fields
Prime Models
Omitting Types on Fields
Filters and Fields
Filters and Products
Chapter Exercises
Filters and Ultraproducts on Projective Sets
General Definitions
Generic Functors and Language String Models
Functorial Fragment Consistency
Filters
Structure Products
Completing Theories and Fragments
Prime Models and Model Completion
Uniform and countably incomplete ultrafilters
Functorial Projetive Set Models and Saturation
Ultraproducts and Ultrafliters
Chapter Exercises
A Glimpse on m Algebraic Set Theory
Preliminaries
Ultraproducts and Ultrafilters on Sets
Ultrafilters over N
Saturation and Preservations
Functorial Models and Descriptive Sets
Filters, Fragment Constructible Models, and Sets
Index
Biography
Dr. Cyrus F. Nourani is a consultant in computing R&D and a research professor at Simon Fraser University. He has many years of experience in the design and implementation of computing systems and has authored/coauthored several books and over 350 publications in mathematics and computer science. He has also held faculty positions at numerous institutions, including the University of Michigan, University of Pennsylvania, University of Auckland, UCLA, and MIT. His research interests include computer science, artificial intelligence, mathematics, virtual haptic computation, information technology, and management.
